Krantz W.B. Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation

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CONCEPT OF THE MICROSCALE ELEMENT 363

Gas in Liquid out

Gas out

Microscale element

Macroscale element

Δz

Liquid

Gas bubble

, c

ˆˆ

ΔS

Infinitesimal

convected

element

Liquid in

Figure 6.2-1 Bubble column for countercurrent gas absorption; the gas diffuses into the

liquid, where it is chemisorbed; the microscale element is a gas bubble; the ﬂux of the soluble

solute is controlled on the microscale by a thin region of inﬂuence within the surrounding

liquid having thickness δ

, as shown in the enlarged view; the macroscale element is an

incremental slice z of the entire absorption column.

The microscale element concept can be grasped most easily by means of an

example. Consider chemisorption

in a bubble column that involves the absorption of

a component from gas bubbles moving upward through a downward-ﬂowing liquid,

as shown in Figure 6.2-1. Chemisorption is used to increase both the amount that

can be transferred and the mass-transfer rates, thereby allowing the use of a smaller

and less costly column to accomplish the same absorption. The microscale element

in this case is a gas bubble. The mass transfer of the soluble component from this

bubble occurs on the microscale within a thin region of inﬂuence or boundary layer in

the liquid whose thickness is δ

. This thickness is controlled by the hydrodynamics;

that is, the inﬂuence of the gas–liquid interface on the ﬂow will create a relatively

stagnant region near the gas bubble wherein the mass transfer on the microscale

will be conﬁned. The concentration gradients of the diffusing components will occur

over the entire thickness δ

for sufﬁciently long contact times and slow reaction

rates. However, for short contact times or sufﬁciently fast reaction rates, they will

occur over shorter length scales, identiﬁed in Section 6.3 using scaling analysis.

An upper bound on δ

is provided by the mass-transfer coefﬁcient k

•

for purely

macroscale equations. However, both conventions for deﬁning the microscale element lead to the same

ﬁnal results for describing the various mass-transfer regimes.

Chemisorption refers to a component going into solution by a chemical potential driving force

accompanied by chemical reaction. In contrast, physical absorption involves only a chemical potential

driving force that leads to a thermodynamic equilibrium concentration of the transferring component

for a sufﬁciently long contact time.

364 APPLICATIONS IN MASS TRANSFER WITH CHEMICAL REACTION

diffusive mass transfer in the absence of chemical reaction as shown in Section 6.3.

The contact time between the gas and liquid that is available for mass transfer on

the microscale can be estimated from the size of a bubble and the relative velocity

between the gas and liquid. Depending on the contact time, a simple model can be

developed to describe the mass-transfer ﬂux from the gas to the liquid. Whether this

simple model needs to include the effects of chemical reaction on the consumption

of the reactants and steepening of the concentration proﬁles depends on the reaction

time scale relative to the characteristic diffusion time. In any event, the mass-transfer

ﬂux occurring on the microscale of a bubble is considered to occur at a point on the

macroscale of the gas-absorption column; that is, one converts the mass-transfer ﬂux

per unit area of the microscale element into a species-generation rate per unit volume

of contacting device in the species-balance equations for the absorbing component

by multiplying the former by the packing area per unit volume of column. One then

integrates the species-balance equation for the liquid phase over the length of the

absorption column to determine its overall performance.

6.3 SCALING THE MICROSCALE ELEMENT

As an example of scaling the describing equations for mass transfer from a micro-

scale element, we consider the chemisorption of component A from a gas that is

bubbled upward through a liquid consisting initially of a nonreacting nonvolatile

solvent S and a reacting nonvolatile solute B that is ﬂowing downward in a vertical

column as shown in Figure 6.2-1. Although a chemisorption process is considered

here, the methodology outlined for the scaling analysis is general and can be applied

to other microscale elements involving mass transfer with chemical reaction as well

as to other systems, such as phase-transition phenomena that can be described by

microscale–macroscale modeling.

In this example we assume irreversible kinetics involving reactants A and B but

will not specify the form of the reaction-rate equation, in order to keep the results

as general as possible. Irreversible kinetics means that the reaction can proceed

in only one direction; that is, the products formed by the reaction of A and B

cannot decompose to reform A and B. We will assume that component A is the

limiting reactant; that is, its concentration limits the amount of reaction that can

occur. For chemisorption the component being absorbed is usually the limiting

reactant; that is, the concentration of component B is maintained at a sufﬁciently

high level to permit the maximum possible absorption of component A in the bulk

liquid. However, for an instantaneous reaction it is possible that component B will

become the limiting reactant at least within some region on the microscale. This

is explored further in Section 6.7.

Solute transfer from the microscale element consisting of the gas bubble will be

controlled by mass transfer through a region of inﬂuence having thickness δ

.The

system considered for developing the describing equations will be an inﬁnitesimal

element of liquid having area S being convected through the region of inﬂuence

as shown by the dotted lines in the enlarged view in Figure 6.2-1. We describe the

SCALING THE MICROSCALE ELEMENT 365

mass transfer within this system in a coordinate system convected at the local rela-

tive velocity between the liquid and the gas bubble. We ignore both transverse and

streamwise convective mass transfer within the convected inﬁnitesimal element;

recall from Chapter 5 that these approximations are justiﬁed for dilute solutions of

the diffusing species and for small solutal Peclet numbers; the latter will be small,

owing to the small characteristic length for the thickness of the region of inﬂu-

ence. We also ignore both streamwise diffusion and curvature effects on the mass

transfer; we saw in Chapter 5 that these approximations are justiﬁed if δ

/L  1,

in which L is the characteristic streamwise length, which is the diameter of the

gas bubble in this example. These assumptions are not limiting in practice since

these neglected effects are accounted for, at least in part, by using mass-transfer

coefﬁcient correlations for this contacting device in order to determine the thick-

ness of the region of inﬂuence. The dilute solution assumption implies that in the

absence of chemical reaction on the microscale, there will be diffusion only of

component A since B will not have a concentration gradient. We also assume that

the mass transfer is liquid-phase controlled; that is, any resistance to mass trans-

fer due to the diffusion of component A in the gas phase is negligible. However,

in Section 6.7 we consider instantaneous reaction conditions for which the mass

transfer can become gas-phase controlled.

Mass transfer from the microscale element to the convected inﬁnitesimal element

can occur only during this characteristic time that they are in contact. Hence, the

appropriate contact time for the unsteady-state mass transfer from the microscale

element to the liquid is the time required for the convected inﬁnitesimal element

to pass around a gas bubble. Once this convected inﬁnitesimal element passes over

the gas bubble, it is assumed to mix intimately with the bulk liquid. The microscale

model will provide the continuous mass-transfer ﬂux of solute from the gas to the

liquid phase, which then will be converted into a homogeneous source term in the

species balance on the macroscale. This mass-transfer ﬂux can inﬂuence the local

concentration of both reactants on the macroscale of the absorption column, for

which both reactants can continue to undergo convective diffusion and reaction.

In view of the considerations above, the resulting describing equations in general

will involve unsteady-state one-dimensional diffusion and homogeneous chemical

reaction. Hence, equation (G.1-5) in the Appendices simpliﬁes to the following for

the two transferring components (step 1):

∂c

∂t

+ u

∂c

∂x

= D

∂

∂y

+ R

(6.3-1)

∂c

∂t

+ u

∂c

∂x

= D

∂

∂y

+ κR

(6.3-2)

where c

and c

denote the molar concentrations of components A and B in

the liquid within the microscale element, u

is the liquid mass-average velocity

relative to a coordinate system located on the gas bubble, y is a coordinate nor-

mal to the surface of the gas bubble as shown in Figure 6.2-1, D

and D

are

the effective binary diffusion coefﬁcients of components A and B in the liquid S

366 APPLICATIONS IN MASS TRANSFER WITH CHEMICAL REACTION

that are assumed constant for the dilute solutions being considered here, κ is a

stoichiometric coefﬁcient that relates how many moles of component B are con-

sumed per mole of A,andR

is the rate of production (moles/volume ·time) of

component A by the homogeneous chemical reaction; note that for chemisorption,

< 0. Note that each term in equation (G.1-5) has been divided by the molec-

ular weight of component A or B in arriving at equations (6.3-1) and (6.3-2). It

is convenient to transform to a coordinate system that is convected at the liquid

velocity relative to the gas bubble, which will be assumed to be constant; that is,

equations (6.3-1) and (6.3-2) are transformed to a moving coordinate system for

which ˜x ≡ x − u

t,inwhichu

is assumed to be the average x-component of the

velocity in the region of inﬂuence around the gas bubble. Equations (6.3-1) and

(6.3-2) assume the following form in this convected coordinate system:

∂c

∂t

= D

∂

∂y

+ R

(6.3-3)

∂c

∂t

= D

∂

∂y

+ κR

(6.3-4)

in which the time derivatives are now evaluated at a constant value of ˜x rather

than at a constant value of x.

The initial and boundary conditions for equations (6.3-3) and (6.3-4) are

given by

=ˆc

at t = 0 (6.3-5)

= c

◦

∂c

∂y

= 0aty = 0 (6.3-6)

=ˆc

at y = δ

(6.3-7)

where ˆc

and ˆc

are the molar concentrations of components A and B in the

bulk liquid on the macroscale and c

◦

is the equilibrium molar concentration of

component A in the liquid at the interface of the gas bubble. Equation (6.3-5)

states that the initial concentrations at the upstream leading edge of the microscale

element are those of the local bulk liquid, as shown in Figure 6.2-1; this translates

to an initial condition for the convected inﬁnitesimal element being considered

here. Although the liquid entering the absorption column is assumed to consist

of only the nonreacting solvent S and reacting solute B, ˆc

can be nonzero and

ˆc

can be less than its inlet concentration, due to the transfer of component A

to the bulk liquid and depletion of component B by chemical reaction that occur

due to upstream microscale elements; that is, these are local bulk concentrations

that can change with axial position on the macroscale of the absorption column.

Equation (6.3-6) states that thermodynamic equilibrium prevails at the gas–liquid

interface for component A and that component B is insoluble in the gas phase; that

is, it has no mass ﬂux into the gas phase. Equation (6.3-7) states that the undisturbed

bulk liquid concentrations prevail at the edge of the region of inﬂuence as shown

in Figure 6.2-1.

SCALING THE MICROSCALE ELEMENT 367

Introduce the following dimensionless variables containing unspeciﬁed scale and

reference factors (steps 2, 3, and 4):

∗

≡

− c

; c

∗

≡

− c

; R

∗

≡

; y

∗

≡

; t

∗

≡

(6.3-8)

We have introduced reference factors for the concentrations of both components

since neither is naturally referenced to zero. Substitute these dimensionless variables

into the describing equations and divide through by the dimensional coefﬁcient of

one term in each equation (steps 5 and 6):

∂c

∗

∂t

∗

∂

∗

∂y

∗2

∗

(6.3-9)

∂c

∗

∂t

∗

∂

∗

∂y

∗2

κR

∗

(6.3-10)

∗

ˆc

− c

∗

ˆc

− c

at t

∗

= 0 (6.3-11)

∗

◦

− c

∂c

∗

∂y

∗

= 0aty

∗

= 0 (6.3-12)

∗

ˆc

− c

∗

ˆc

− c

at y

∗

(6.3-13)

We divided through by the coefﬁcient of the diffusion term in equations

(6.3-9) and (6.3-10) because it provides the only means by which mass can get

into the convected inﬁnitesimal element to react or accumulate and hence must be

retained.

We now set appropriate dimensionless groups equal to zero or 1 to deter-

mine the reference and scale factors, respectively (step 7). If we assume that the

reaction and diffusion of both components occur over the entire thickness of the

liquid layer on the microscale, the appropriate length scale is obtained by setting

= 1 to obtain y

= δ

. The scale factor for the reaction rate is chosen to be

some characteristic maximum value R

. If the form of the reaction-rate equation

were speciﬁed, R

would be determined from the reaction-rate constant and rel-

evant concentrations. We can reference c

∗

to zero by setting ( ˆc

− c

)/c

= 0

to obtain c

=ˆc

since ˆc

is its smallest possible value. We can bound c

∗

◦

(1) by setting (c

◦

− c

)/c

= (c

◦

−ˆc

)/c

= 1toobtainc

= c

◦

−ˆc

which is the driving force for the diffusion of component A. We can bound c

∗

to be

◦

(1) by setting ( ˆc

− c

)/c

= 1toobtainc

=ˆc

− c

. The reference

factor c

is obtained by recognizing that component B will have a concentra-

tion gradient only if there is signiﬁcant chemical reaction on the microscale. This

implies that the reaction term in equation (6.3-10) must be of the same order as

the diffusion term; hence, we set κR

= κR

( ˆc

− c

) = 1

368 APPLICATIONS IN MASS TRANSFER WITH CHEMICAL REACTION

to obtain c

=ˆc

− κR

, which provides us with an estimate of the min-

imum concentration of component B. When these scale and reference factors are

substituted into the describing equations, we obtain

∂c

∗

∂t

∗

∂

∗

∂y

∗2

◦

−ˆc

)

∗

(6.3-14)

∂c

∗

∂t

∗

∂

∗

∂y

∗2

+ R

∗

(6.3-15)

∗

= 0,c

∗

= 1att

∗

= 0 (6.3-16)

∗

= 1,

∂c

∗

∂y

∗

= 0aty

∗

= 0 (6.3-17)

∗

= 0,c

∗

= 1aty

∗

= 1 (6.3-18)

Let us now use the results of our scaling analysis to explore possible simpliﬁ-

cations in the describing equations (step 8). Equation (6.3-14) indicates that there

are three characteristic times for the microscale mass-transfer process:

contact time between the liquid and gas on the microscale (6.3-19)

characteristic time for diffusion of A on the microscale (6.3-20)

◦

−ˆc

characteristic time for reaction on the microscale (6.3-21)

Let us ﬁrst consider the implications of the contact time relative to the charac-

teristic time for diffusion. Note that the characteristic time for diffusion must be at

least of the same order as the contact time if any mass transfer is to occur, since

diffusion is the only transport mechanism available to transfer component A from

the gas to the liquid phase. Steady-state mass transfer will apply on the microscale

if the following criterion is satisﬁed:

 1 ⇒ steady-state mass transfer on the microscale (6.3-22)

Equation (6.3-22) implies that the characteristic time for diffusion is much shorter

than the contact time. When this criterion is satisﬁed, the diffusion can penetrate

the entire thickness of the liquid region δ

within the time that the liquid ﬂows

over the gas bubble. Hence, the mass transfer on the microscale can be described

by a ﬁlm theory model, for the reasons discussed in Section 5.2.

Let us use the results of our scaling analysis for steady-state mass transfer

to establish a general relationship between k

•

, the mass-transfer coefﬁcient for

purely physical absorption, and δ

, the effective thickness of the ﬂuid layer for

SCALING THE MICROSCALE ELEMENT 369

the microscale element. Let us scale the equation that deﬁnes the mass-transfer

coefﬁcient, which is given by

•

≡

− x

+ N

)]|

y=0

◦

−ˆc

∼

−D

(∂c

/∂y)

y=0

◦

−ˆc

(6.3-23)

This equation for the mass-transfer coefﬁcient has been simpliﬁed consistent with

the dilute solution assumption, for which the molar density is approximately con-

stant. Introduce the scale factors obtained from our scaling analysis for steady-state

mass transfer on the microscale into equation (6.3-23):

•

=−

∂c

∗

∂y

∗



∗

(6.3-24)

Since ∂c

∗

/∂y

∗

=(1), it follows that for steady-state purely physical absorp-

tion,

•

steady-state physical absorption (6.3-25)

where k

•

denotes the value of k

•

for purely physical absorption. It is determined

solely by the hydrodynamics for purely physical absorption in the absence of any

chemical reaction. Equation (6.3-25) then establishes that the mass-transfer coefﬁ-

cient for purely physical absorption can be used to estimate the maximum diffusive

penetration distance δ

, which was stated without proof in Section 6.2.

Consider now the case when the contact time is very short. This causes a contra-

diction in equation (6.3-14) since the diffusion term must balance the unsteady-state

term; that is, component A can accumulate in the liquid solution only by diffus-

ing into it from the gas phase. The forgiving nature of scaling indicates that δ

is not the proper characteristic length scale for short contact times; that is, there

is a region of inﬂuence or solutal boundary layer δ

within which component A

experiences a characteristic change in concentration. The thickness of this solutal

boundary layer is determined by setting the dimensionless group multiplying the

unsteady-state term in equation (6.3-9) equal to 1, thereby obtaining



(6.3-26)

Choosing this characteristic length ensures that both the diffusion and unsteady-

state terms are retained in equation (6.3-9). If δ

 δ

, the boundary condition on

component A given in equation (6.3-13) can be applied at inﬁnity and a penetration

theory model can be developed to describe the mass transfer with chemical reac-

tion on the microscale, for the reasons discussed in Section 5.3. This penetration

boundary-layer approximation can be made when the following criterion is satisﬁed:

1 

⇒ unsteady-state penetration theory applies (6.3-27)

where Fo

≡ D

/δ

is the solutal Fourier number.

370 APPLICATIONS IN MASS TRANSFER WITH CHEMICAL REACTION

Let us use the results of our scaling analysis for unsteady-state mass transfer

to establish the relationship between k

•

, the mass-transfer coefﬁcient for purely

physical absorption, and t

, the contact time between the liquid and the gas for the

microscale element. Introduce the scale factors obtained from our scaling analysis

for unsteady-state mass transfer in the microscale element into equation (6.3-23):

•

√

= k

•



=−

∂c

∗

∂y

∗



∗

(6.3-28)

Since ∂c

∗

/∂y

∗

=(1), it follows that for unsteady-state purely physical

absorption

•



unsteady-state physical absorption (6.3-29)

where k

•

again denotes the value of k

•

for purely physical absorption, which is

determined solely by the hydrodynamics.

In the following sections we consider the implications of the reaction time

scale relative to that for diffusion. The mass transfer from the microscale to the

macroscale can be markedly different depending on the relative magnitude of these

two time scales. Figure 6.3-1 shows the concentration proﬁles of components A

and B for the assumed irreversible chemical reaction for ﬁve characteristic reac-

tion regimes that can occur in the microscale element, depending on the magnitude

of the reaction time scale relative to that for diffusion for component A. Scaling

analysis will now be used to determine the conditions for these various reaction

regimes.

slow reaction regime

or c

intermediate reaction regime

fast reaction regime

inner domain of the

instantaneous reaction regime

surface domain of the

instantaneous reaction regime

Figure 6.3-1 Representative concentration proﬁles for components A and B as a function

of distance within the liquid layer of the microscale element for chemisorption with an

irreversible reaction.

INTERMEDIATE REACTION REGIME 371

6.4 SLOW REACTION REGIME

The characteristic time for reaction can be much longer than that for diffusion, in

which case no chemical reaction will occur on the microscale, thereby justifying

ignoring the reaction term in equation (6.3-14). This is referred to as the slow

reaction regime approximation. Equation (6.3-14) indicates that this approximation

is applicable when the following criterion is satisﬁed:

◦

−ˆc

)

 1 ⇒ slow reaction regime (6.4-1)

The criterion above involves the characteristic diffusion time for component A

rather than B because A must diffuse from the gas to the liquid phase for any

reaction to occur, irrespective of how rapidly B can diffuse from the bulk liquid to

the microscale element. Note that the slow reaction regime implies that c

=ˆc

−

κR

∼

ˆc

; that is, it implies that there is no change in the concentration

of component B on the microscale. Figure 6.3-1 shows that for the slow reaction

regime the concentration gradient of component A occurs over the entire thickness

, whereas there is no concentration gradient for component B, owing to the

absence of any chemical reaction; the linear concentration proﬁle for component

A for the slow reaction regime shown in Figure 6.3-1 is for the special case of

steady-state diffusion corresponding to a long contact time. For short contact times,

the concentration proﬁle of component A would be nonlinear and extend over a

thickness δ

<δ

. Note that in the slow reaction regime, the bulk concentration of

component A can range from 0 ≤ˆc

≤ c

◦

for an irreversible reaction, depending

on how fast the reaction is on the macroscale relative to mass transfer from the

microscale element. The slow reaction regime implies that correlations for purely

physical absorption in the absence of chemical reaction can be used to describe the

mass transfer of component A from the microscale to the macroscale. Note that

the slow reaction regime approximation implies only that chemical reaction does

not inﬂuence the mass transfer on the microscale; it could inﬂuence the transport

on the macroscale, as we will see in Section 6.8.

6.5 INTERMEDIATE REACTION REGIME

If the characteristic time for reaction is of the same order as that for diffusion,

the reaction term must be retained in equation (6.3-14). This will accelerate the

absorption of component A from that for purely physical absorption. Moreover, it

will cause some depletion and thereby diffusion of component B. We refer to this

as the intermediate reaction regime approximation

; it applies when the following

criterion is satisﬁed:

◦

−ˆc

)

◦

(1) ⇒ intermediate reaction regime (6.5-1)

The intermediate reaction regime deﬁned here is sometimes referred to as the transition between the

slow to fast reaction regimes.

372 APPLICATIONS IN MASS TRANSFER WITH CHEMICAL REACTION

The minimum concentration of component B due to the chemical reaction can be

estimated from the reference factor that we determined for this component from

our scaling analysis; that is,

=ˆc

−

κR

(6.5-2)

Figure 6.3-1 shows that in the intermediate reaction regime, components A and B

both undergo a characteristic change in concentration over the length δ

. However,

the reaction is not fast enough to reduce the concentration of either component to

zero within the microscale element. In the intermediate reaction regime, the bulk

concentration of component A can range from 0 ≤ˆc

◦

for an irreversible

reaction, depending on the time scale for reaction on the macroscale relative to the

time scale for mass transfer from the microscale element. The mass-transfer ﬂux in

the intermediate reaction regime is larger than that for the slow reaction regime, as

shown by the increase in the slope of the concentration proﬁle for component A at

y = 0. It would appear from equation (6.5-2) that the concentration of component

B could be reduced to zero for sufﬁciently fast reaction rates. However, this is not

possible in the intermediate reaction regime since the concentration of component

A would have to be reduced to zero ﬁrst because it is the limiting reactant. The

latter can occur within the microscale element only in the fast or instantaneous

reaction regimes.

6.6 FAST REACTION REGIME

Consider now conditions for which the characteristic time for reaction is much

shorter than the characteristic time for diffusion. This is referred to as the fast

reaction regime, which applies when the following criterion is satisﬁed:

◦

−ˆc

)

 1 ⇒ fast reaction regime (6.6-1)

For these conditions the forgiving nature of scaling indicates a contradiction in

equation (6.3-14); that is, the reaction term no longer balances the diffusion term

that must be retained in order to satisfy the boundary conditions. This implies that

the length scale y

= δ

is not appropriate for this reaction regime and suggests

that there is a region of inﬂuence or reaction boundary layer δ

within which the

irreversible reaction of component A goes to completion at which ˆc

= 0. The

thickness of this reaction boundary layer is obtained by setting the dimensionless

group that multiplies the reaction term in equation (6.3-14) equal to 1 to obtain



◦



1/2

(6.6-2)

Figure 6.3-1 shows that in the fast reaction regime the concentration of compo-

nent A is reduced to zero within the distance δ

. However, since component A is